Volatility and dividend risk in perpetual American options

American options are financial instruments that can be exercised at any time before expiration. In this paper we study the problem of pricing this kind of derivatives within a framework in which some of the properties --volatility and dividend policy-- of the underlaying stock can change at a random instant of time, but in such a way that we can forecast their final values. Under this assumption we can model actual market conditions because some of the most relevant facts that may potentially affect a firm will entail sharp predictable effects. We will analyse the consequences of this potential risk on perpetual American derivatives, a topic connected with a wide class of recurrent problems in physics: holders of American options must look for the fair price and the optimal exercise strategy at once, a typical question of free absorbing boundaries. We present explicit solutions to the most common contract specifications and derive analytical expressions concerning the mean and higher moments of the exercise time.Comment: 21 pages, 5 figures, iopart, submitted for publication; deep revision, two new appendice

Entanglements in Quiescent and Sheared Polymer Melts

We visualize entanglements in polymer melts using molecular dynamics simulation. A bead at an entanglement interacts persistently for long times with the non-bonded beads (those excluding the adjacent ones in the same chain). The interaction energy of each bead with the non-bonded beads is averaged over a time interval $\tau$ much longer than microscopic times but shorter than the onset time of tube constraints $\tau_{\rm e}$. Entanglements can then be detected as hot spots consisting of several beads with relatively large values of the time-averaged interaction energy. We next apply a shear flow with rate much faster than the entangle motion. With increasing strain the chains take zigzag shapes and a half of the hot spots become bent. The chains are first stretched as a network but, as the bends approach the chain ends, disentanglements subsequently occur, leading to stress overshoot observed experimentally.Comment: 19 pages, 11 figure

Theoretical description of a DNA-linked nanoparticle self-assembly

Nanoparticles tethered with DNA strands are promising building blocks for bottom-up nanotechnology, and a theoretical understanding is important for future development. Here we build on approaches developed in polymer physics to provide theoretical descriptions for the equilibrium clustering and dynamics, as well as the self-assembly kinetics of DNA-linked nanoparticles. Striking agreement is observed between the theory and molecular modeling of DNA tethered nanoparticles.Comment: Accepted for publication in Physical Review Letter

Thermal Fluctuations and Rubber Elasticity

The effects of thermal elastic fluctuations in rubber materials are examined. It is shown that, due to an interplay with the incompressibility constraint, these fluctuations qualitatively modify the large-deformation stress-strain relation, compared to that of classical rubber elasticity. To leading order, this mechanism provides a simple and generic explanation for the peak structure of Mooney-Rivlin stress-strain relation, and shows a good agreement with experiments. It also leads to the prediction of a phonon correlation function that depends on the external deformation.Comment: 4 RevTeX pages, 1 figure, submitted to PR

Skating on a Film of Air: Drops Impacting on a Surface

Drops impacting on a surface are ubiquitous in our everyday experience. This impact is understood within a commonly accepted hydrodynamic picture: it is initiated by a rapid shock and a subsequent ejection of a sheet leading to beautiful splashing patterns. However, this picture ignores the essential role of the air that is trapped between the impacting drop and the surface. Here we describe a new imaging modality that is sensitive to the behavior right at the surface. We show that a very thin film of air, only a few tens of nanometers thick, remains trapped between the falling drop and the surface as the drop spreads. The thin film of air serves to lubricate the drop enabling the fluid to skate on the air film laterally outward at surprisingly high velocities, consistent with theoretical predictions. Eventually this thin film of air must break down as the fluid wets the surface. We suggest that this occurs in a spinodal-like fashion, and causes a very rapid spreading of a wetting front outwards; simultaneously the wetting fluid spreads inward much more slowly, trapping a bubble of air within the drop. Our results show that the dynamics of impacting drops are much more complex than previously thought and exhibit a rich array of unexpected phenomena that require rethinking classical paradigms.Comment: 4 pages, 4 figure

The Hydrodynamic Interaction in Polymer Solutions Simulated with Dissipative Particle Dynamics

We analyzed extensively the dynamics of polymer chains in solutions simulated with dissipative particle dynamics (DPD), with a special focus on the potential influence of a low Schmidt number of a typical DPD fluid on the simulated polymer dynamics. It has been argued that a low Schmidt number in a DPD fluid can lead to underdevelopment of the hydrodynamic interaction in polymer solutions. Our analyses reveal that equilibrium polymer dynamics in dilute solution, under a typical DPD simulation conditions, obey the Zimm model very well. With a further reduction in the Schmidt number, a deviation from the Zimm model to the Rouse model is observed. This implies that the hydrodynamic interaction between monomers is reasonably developed under typical conditions of a DPD simulation. Only when the Schmidt number is further reduced, the hydrodynamic interaction within the chains becomes underdeveloped. The screening of the hydrodynamic interaction and the excluded volume interaction as the polymer volume fraction is increased are well reproduced by the DPD simulations. The use of soft interaction between polymer beads and a low Schmidt number do not produce noticeable problems for the simulated dynamics at high concentrations, except that the entanglement effect which is not captured in the simulations.Comment: 27 pages, 13 page

Autocorrelation of Random Matrix Polynomials

We calculate the autocorrelation functions (or shifted moments) of the characteristic polynomials of matrices drawn uniformly with respect to Haar measure from the groups U(N), O(2N) and USp(2N). In each case the result can be expressed in three equivalent forms: as a determinant sum (and hence in terms of symmetric polynomials), as a combinatorial sum, and as a multiple contour integral. These formulae are analogous to those previously obtained for the Gaussian ensembles of Random Matrix Theory, but in this case are identities for any size of matrix, rather than large-matrix asymptotic approximations. They also mirror exactly autocorrelation formulae conjectured to hold for L-functions in a companion paper. This then provides further evidence in support of the connection between Random Matrix Theory and the theory of L-functions

Theoretical and numerical study of the phase diagram of patchy colloids: ordered and disordered patch arrangements

We report theoretical and numerical evaluations of the phase diagram for a model of patchy particles. Specifically we study hard-spheres whose surface is decorated by a small number f of identical sites ("sticky spots'') interacting via a short-range square-well attraction. We theoretically evaluate, solving the Wertheim theory, the location of the critical point and the gas-liquid coexistence line for several values of f and compare them to results of Gibbs and Grand Canonical Monte Carlo simulations. We study both ordered and disordered arrangements of the sites on the hard-sphere surface and confirm that patchiness has a strong effect on the phase diagram: the gas-liquid coexistence region in the temperature-density plane is significantly reduced as f decreases. We also theoretically evaluate the locus of specific heat maxima and the percolation line.Comment: preprint, 32 pages, 6 figures, 3 tables, J. Chem. Phys. in pres

Topological analysis of polymeric melts: Chain length effects and fast-converging estimators for entanglement length

Primitive path analyses of entanglements are performed over a wide range of chain lengths for both bead spring and atomistic polyethylene polymer melts. Estimators for the entanglement length N_e which operate on results for a single chain length N are shown to produce systematic O(1/N) errors. The mathematical roots of these errors are identified as (a) treating chain ends as entanglements and (b) neglecting non-Gaussian corrections to chain and primitive path dimensions. The prefactors for the O(1/N) errors may be large; in general their magnitude depends both on the polymer model and the method used to obtain primitive paths. We propose, derive and test new estimators which eliminate these systematic errors using information obtainable from the variation of entanglement characteristics with chain length. The new estimators produce accurate results for N_e from marginally entangled systems. Formulas based on direct enumeration of entanglements appear to converge faster and are simpler to apply.Comment: Major revisions. Developed near-ideal estimators which operate on multiple chain lengths. Now test these on two very different model polymers

Scaling of Entropic Shear Rigidity

The scaling of the shear modulus near the gelation/vulcanization transition is explored heuristically and analytically. It is found that in a dense melt the effective chains of the infinite cluster have sizes that scale sub-linearly with their contour length. Consequently, each contributes k_B T to the rigidity, which leads to a shear modulus exponent d\nu. In contrast, in phantom elastic networks the scaling is linear in the contour length, yielding an exponent identical to that of the random resistor network conductivity, as predicted by de Gennes'. For non-dense systems, the exponent should cross over to d\nu when the percolation length becomes much larger than the density-fluctuation length.Comment: 4 pages, 2 eps figure